It really depends on the title. Honestly, an 8320 pairs pretty well with a 960 most of the time. A 960 isn't exactly a powerhouse GPU.
From a CPU perspective, anything remotely clsoe to 'best' doesn't come at a $100-$220 pricepoint..... especially considering you'll need a new motherboard and DDR4 memory.
An 8370, 9590 etc are socket compatible, but not a huge upgrade (8370 and just a nightmare to get stable, while achieving oretty poor performance (9590).
If you're set on an upgrade, from an 8320 I'd look to no less than an i5-6XXX or 7XXXX, or a Ryzen5
Absolute minimum, those upgrades are going to set you back ~$260
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 1400 3.2GHz Quad-Core Processor ($158.69 @ OutletPC)
Motherboard: ASRock - A320M-DGS Micro ATX AM4 Motherboard ($54.98 @ Newegg)
Memory: Patriot - Signature Line 8GB (2 x 4GB) DDR4-2400 Memory ($57.99 @ Newegg)
Total: $271.66
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-07 17:42 EDT-0400
But that limits your RAM to 8GB, and a non-OCing board (With a CPU that really should be OC'd).
From an Intel standpoint, you're looking closer to $300
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i5-7400 3.0GHz Quad-Core Processor ($175.88 @ OutletPC)
Motherboard: MSI - B250M PRO-VD Micro ATX LGA1151 Motherboard ($62.88 @ OutletPC)
Memory: Patriot - Signature Line 8GB (2 x 4GB) DDR4-2400 Memory ($57.99 @ Newegg)
Total: $296.75
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-07 17:44 EDT-0400
And that's a CPU that can't OC, and again, 8GB RAM.
I'd look to Ryzen, but budget more for in the $300-$400 range
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor ($209.99 @ Amazon)
Motherboard: ASRock - AB350M-HDV Micro ATX AM4 Motherboard ($67.98 @ Newegg)
Memory: Team - Dark 16GB (2 x 8GB) DDR4-3000 Memory ($104.99 @ Newegg)
Total: $382.96
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-07 17:45 EDT-0400
Your 'problem' then becomes the 960. Ryzen5 are capable of nice pairings with GPUs mcuh, much greater than a 960